Invariants and K-spectrums of local theta lifts

TL;DR

This paper investigates the properties of local theta lifts in the stable range for reductive dual pairs, establishing conditions for irreducibility, unitarity, and describing their geometric invariants and spectra.

Contribution

It proves that under certain conditions, the full local theta lift coincides with the classical lift, and describes their associated varieties, cycles, and spectra in terms of nilpotent orbits.

Findings

Full theta lift equals the classical theta lift except in one case.

Associated varieties and cycles of theta lifts are related to those of the original representation.

Constructs modules with spectra isomorphic to sections over nilpotent orbits.

Abstract

Let $(G,G')$ be a type I irreducible reductive dual pair in $\mathrm{Sp}(W_{\mathbb{R}})$. We assume that $(G,G')$ is in the stable range where $G$ is the smaller member. Let $K$ and $K'$ be maximal compact subgroups of $G$ and $G'$ respectively. Let $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ and $\mathfrak{g}' = \mathfrak{k}' \oplus \mathfrak{p}'$ be the complexified Cartan decompositions of the Lie algebras of $G$ and $G'$ respectively. Let ${\widetilde{K}}$ and ${\widetilde{K}}'$ be the inverse images of $K$ and $K'$ in the metaplectic double cover $\widetilde{\mathrm{Sp}}(W_\mathbb{R})$ of ${\mathrm{Sp}}(W_\mathbb{R})$. Let $\rho$ be a genuine irreducible $(\mathfrak{g},{\widetilde{K}})$-module. Our first main result is that if $\rho$ is unitarizable, then except for one special case, the full local theta lift $\rho' = \Theta(\rho)$ is equal to the local theta lift $\theta(\rho)$. Thus excluding the special case, the full theta lift $\rho'$ is an irreducible and unitarizable $(\mathfrak{g}',{\widetilde{K}}')$-module. Our second main result is that the associated variety and the associated cycle of $\rho'$ are the theta lifts of the associated variety and the associated cycle of the contragredient representation $\rho^*$ respectively. Finally we obtain some interesting $(\mathfrak{g},{\widetilde{K}})$-modules whose ${\widetilde{K}}$-spectrums are isomorphic to the spaces of global sections of some vector bundles on some nilpotent $K_\mathbb{C}$-orbits in $\mathfrak{p}^*$.